Given \(d\)  distinct spheres \(S_1,\cdots ,S_d\)  in \(A\) , one can find a ball containing them by first connecting \(S_{i}\)  to \(S_{i+1}\)  for \(i\le \lbrace 1,\cdots ,d-1\rbrace \)  using \(d-1\)  disjointly embedded arcs, and then taking a small neighborhood of the union of these arcs together with the \(d\)  spheres. Since \(W_g^b\)  is simply connected and has dimension at least 6, these arcs are unique up to isotopy, and can be chosen to be pairwise disjoint. As a consequence, \(\alpha \)  is a surjective simplicial map.
To see that \(\alpha \)  is injective, suppose there are two embedded balls \(B, B^{\prime }\)  each containing \(d\)  spheres \(S_1,\cdots ,S_d\)  in \(A\) . Roughly as above, we choose a set \(d-1\)  arcs \(a_i\)  (resp. \(a_i^{\prime }\) ) in \(B\)  (resp. \(B^{\prime }\) ) connecting \(S_{i}\)  to \(S_{i+1}\)  for \(i\le \lbrace 1,\cdots ,d-1\rbrace \) . Since \(W_g^b\)  is simply-connected, we may isotope \(a_i \)  to \(a_i^{\prime }\) . Applying a further isotopy that shrinks \(B\) , we can assume that \(B\)  lies in the interior of \(B^{\prime }\) . By the \(h\) -cobordism theorem [1]}, \(B\)  and \(B^{\prime }\)  are in fact isotopic inside \(W_g^b\) .
